In this paper, we solve a non-linear reaction–diffusion system with Dirichlet–Neumann mixed boundary conditions using a finite difference method (FDM) in space and the implicit midpoint method in time. This type of system appears, e.g., in the mathematical modeling of light-controlled drug delivery. One of the key results of this paper is the proof that the method has superconvergence second-order in space in a discrete -norm and optimal second-order convergence in time in a discrete -norm. Our result relies on the direct analysis of a suitable error equation, avoiding the classic construction of consistency plus stability implies convergence. One advantage of such an analysis technique is the establishment of the method’s non-linear stability in an elegant way. Numerical examples support the theoretical convergence result.

中文翻译：

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光控药物输送系统的数值分析

在本文中，我们使用空间上的有限差分法（FDM）和时间上的隐式中点法求解具有狄利克雷-诺依曼混合边界条件的非线性反应扩散系统。这种类型的系统出现在例如光控药物输送的数学模型中。本文的关键成果之一是证明了该方法在离散范数下具有空间二阶超收敛性和在离散范数下时间上的最优二阶收敛性。我们的结果依赖于对合适误差方程的直接分析，避免了一致性加稳定性意味着收敛的经典构造。这种分析技术的一个优点是以一种优雅的方式建立该方法的非线性稳定性。数值例子支持了理论收敛结果。